Energy Conversion and Management 53 (2012) 337–345

Contents lists available at ScienceDirect

Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

On-line supercapacitor dynamic models for energy conversion and management

C.H. Wu a, Y.H. Hung b, C.W. Hong a,⇑a Dept. of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwanb Dept. of Industrial Education, National Taiwan Normal University, Taipei, Taiwan

a r t i c l e i n f o

Article history:Received 5 April 2010Received in revised form 13 January 2011Accepted 29 January 2011Available online 23 March 2011

Keywords:SupercapacitorAlternative current impedanceNeural network

0196-8904/$ - see front matter 2011 Elsevier Ltd. Adoi:10.1016/j.enconman.2011.01.018

⇑ Corresponding author. Tel.: +886 3 5742591; fax:E-mail address: cwhong@pme.nthu.edu.tw (C.W. H

a b s t r a c t

This paper develops on-line nonlinear dynamic models of electrochemical supercapacitors which are forenergy conversion and management. Based on the theory of electrochemical impedance spectroscopy,extensive alternative current impedance tests have been conducted to investigate the frequency-domaindynamics of these supercapacitors. A Nyquist diagram is plotted to help establish an equivalent electriccircuit, which is regarded as the first-phase linear model. Two performance-influencing factors, environ-mental temperature and operating voltage, are considered as nonlinear effects. The nonlinear relation-ships among parameters of the capacitances and resistances in the first-phase model are establishedby a multi-layer artificial neural network. The neural parameters are trained using a back-propagationalgorithm by feeding the experimental data bank. Combining the first-phase model and the on-line neu-ral ‘‘parameter identifier’’, the algorithm produces an on-line nonlinear dynamic model. Simulationresults have proved that this proposed model is able to achieve both system fidelity and computationalefficiency.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

Supercapacitors (SCs) have recently offered an alternative ap-proach for electrical energy conversion and management systems.They are widely applied in industrial applications, including unin-terruptable power supplies, electric vehicles, windmills, and evenpersonal mobile phones [1]. Supercapacitors are characterized bya high charging/discharging efficiency (up to 95%), fast chargingtime (103–106 s), long cycle life (over 500,000 cycles), relativelyhigh specific power (100,000 W kg1), wide operating temperaturerange (2550 C), etc. Due to these impressive characteristics, asupercapacitor can be employed as a peak power auxiliary deviceto compensate for low power output from an energy source withhigh energy density (such as fuel cells and lithium batteries) [2,3].

The superior characteristics of supercapacitors make them apromising candidate for future power sources. A dynamic model-ing technique for the power conversion is important due to theindustrial requirements of rapid system design for prototype con-trollers and control strategy verifications [4]. Advanced modelingtechnique and numerical schemes are still under development.Performance-based micro-scale modeling and dynamic macro-scale modeling are two main categories of these techniques forsupercapacitors. Micro-scale modeling techniques focus on the ba-sic electrochemical process and look insight the double-layer

ll rights reserved.

+886 3 5722840.ong).

effects, porous electrode materials, geometric patterns, and corre-sponding electrolytes in detail. Dynamic modeling techniques, onthe other hand, regard the supercapacitors as a lumped system intime and frequency domains. Researchers have proposed two dy-namic approaches in recent years. They are based on the analysisof either (i) equivalent electric circuits or (ii) artificial neural net-works (ANNs).

In 2004, Spyker and Nelms proposed a classical equivalent cir-cuit for method (i) that consists of a capacitor, an equivalent seriesresistance, and an equivalent parallel resistance [5]. Parameterswere determined by charging/discharging performance. Gualouset al. employed electrochemical impedance spectroscopy (EIS) toconstruct a second-order equivalent circuit for supercapacitors,and established the temperature effect using a polynomial equa-tion [6]. Buller et al. also employed the EIS method to model thedynamic behavior of a SC in the frequency domain [7]. They sug-gested a third-order transfer function and correlated parameters.In 2006, Michel again used the EIS method to establish a modeldescribing the influencing factors by both open circuit voltage(OCV) and operation temperature of the SC [8]. Lajnef et al. in2007 proposed an equivalent circuit for a peak-powered SC. Theyinvestigated the relationships among the OCV, temperature, andcharging frequency [9]. Also in 2007, Rafik et al. established a 14RLC equivalent circuit to describe the influences of operating fre-quency, voltage, and thermal effects on SCs [10]. In 2008, Shi andCrow proposed two equivalent circuits based on the model derivedfrom [8] as shown in [11]. In 2009, Brouji et al. investigated theinfluence of life cycle on SCs using the EIS method. They

4

5

6

7x 10-3

(Z)

Nyquist Plot of BCAP350 at 300K

Vsc= 0V

Vsc=0.5V

Vsc=1.0V

Vsc=1.5V

Vsc=2.0V

Vsc=2.5V

338 C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345

constructed an electrical equivalent model and proposed variousparameters [12]. In the same year, Sakka et al. built up a thermalmodel for SCs based on their equivalent thermal-electric circuits[13].

For the second category of SC modeling, the ANN technique hasbeen widely used to model highly nonlinear phenomenon in vari-ous industrial and physical fields. By feeding the measured data-base to the input-output ANN layers, system dynamics can beemulated by training ANN neurons through several iterations. Thisapproach regards the modeled system as a black box so that themutual effects of all influencing factors can be simulated and pre-dicted. In 2000, Bhatikar et al. applied ANN for energy storage sys-tem modeling in hybrid electric vehicles [14]. The ANN approachaccurately portrays the complex, nonlinear correlations of the bat-teries. Shen et al. [15] and Monfared et al. [16] modeled the effectsof temperature and charge/discharge current on the state-of-charge (SOC) of batteries using the ANN technique. Although manystudies establish the nonlinear dynamics of the energy storage sys-tem using ANNs, relatively few studies have adopted this approachto model supercapacitors. In 2006, Farsi calculated nonlinear SCperformance using a four-layer ANN with two hidden layers [17].The crystal size, surface lattice length, and exchange currents etc.were the ANN inputs, while utilization ratio, energy density andpower density were the outputs.

This paper combines both techniques of electric equivalent cir-cuits (method 1) and the ANN approach (method 2) to establish anon-line SC model. The proposed approach accommodates the phys-ical meanings and nonlinear dynamics of supercapcitors, and aimsto achieve computational efficiency for their system design. Ahardware-in-the-loop (HIL) platform for a rapid-prototyping SCcontroller can be developed using this newly devised nonlinearmodel.

0.008 0.009 0.01 0.011 0.012 0.013 0.0140

1

2

3

Re (Z)

-Im

Fig. 2. Nyquist plot of the impedance of a supercapacitor measured at 0 V, 0.5 V,1.0 V, 1.5 V, 2.0 V, 2.5 V, respectively.

2. EIS analysis and establishment of nonlinear relationships

The EIS technique has been used to characterize many powersources. This approach makes it possible to determine the influ-ence of physical and chemical phenomena with a single experi-mental procedure encompassing a sufficiently broad range offrequencies. The EIS technique applies a small amplitude sinusoi-dal excitation signal to the system and measures the frequency re-sponse. The Nyquist diagram can be plotted by analyzing the EISdata; allowing the equivalent electric circuit to be identified.

Fig. 1 shows the SC test platform used in this study. The testedSC was first placed in a thermal incubator to regulate the working

Fig. 1. The test platform

temperature. An alternative current (AC) impedance tester thenautomatically fed sinusoidal current commands to the SC undervarious frequencies to derive the EIS results. For charge and dis-charge tests, a PC sent commands via the AD/DA card to a relaycontrol IC board, which switched between the charge and the dis-charge modes. A direct current (DC) power supplier was used tocharge the SC, while a DC electronic load discharged the SC.

Figs. 2 and 3 illustrate the experimental results. To derive thenonlinear relationships, the thermal incubator controlled the oper-ation temperature while the voltages of the tested SC were kept atconstants: 0 V, 0.5 V, 1 V, 2 V, and 2.5 V. The AC impedance ap-proach was then used to analyze the needed information. Fig. 2illustrates the Nyquist plot at 300 K. Each profile intersecting thex-axis indicates the value of the equivalent series resistor (ESR),while the vertical asymptotes along these five profiles representthe serial capacitance and the parallel RC circuit. This figure showthat the value of the ESR increased while VSC decreased. Fig. 3a de-picts the nonlinear relationships of the ESR (denoted by RS) among

for supercapacitors.

270 280 290 300 310 320 3307.5

8

8.5

9

9.5

10

10.5

11 x 10-3

Temperature (K)

Temperature (K)

Rs

(Ohm

ic)

Rs at Different Tempertature & Voltage

Vsc = 0 VVsc = 0.5 VVsc = 1 VVsc = 1.5 VVsc = 2.0 VVsc = 2.5 V

270 280 290 300 310 320 330240

260

280

300

320

340

360

380Capacitance (0.1Hz) at Different Temperature & Voltage

Cap

acita

nce

(F)

Vsc = 0 VVsc = 0.5 VVsc = 1 VVsc = 1.5 VVsc = 2 VVsc = 2.5 V

Fig. 3. Nonlinear relationships of (a) RS and (b) C0 under different workingtemperature and VSC.

8.5 9 9.5 10 10.5 11x 10-3

0

0.5

1

1.5

2

2.5

3 x 10-3

Re (Z)

-Im (Z

)

Nyquist Plot of BCAP350 at 300K

37 Hz

1 Hz

193 mHz

Fig. 4. Nyquist plot of a supercapacitor under various frequencies.

Fig. 5. Equivalent electric circuit and its components of the supercapacitor.

C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345 339

operation temperatures and output voltages. Compared to Fig. 2, itverifies that the ESR grew as the SC voltage decreased. With tem-perature varying from 270 K to 330 K as shown in x-axis, the non-linear relationships of RS(T, VSC) can be constructed. Fig. 3b alsoprovides sufficient information for establishing the capacitancerelationship of the supercapacitor, which can be expressed asCSC = CSC(T, VSC). Note that the charge/discharge frequency was setat 0.1 Hz to develop SCs suitable for (hybrid) electric vehicles.

3. On-line nonlinear SC system model

3.1. Linear SC modeling in frequency domain and in time domain

According to the Nyquist plot described in Section 2, the sup-ercapacitor can be modeled as an equivalent electric circuit. Theexample profile (T = 300 K, VSC = 1.5 V) in Fig. 4 shows that theelectric circuit can be regarded as a circuit with two parallel RC cir-cuits (shown as two half circles). This equivalent circuit is inter-connected in series with a serial RC circuit (shown as one verticalasymptote). Fig. 5 illustrates the equivalent circuit and its compo-nents for modeling a Maxwell BCAP350 supercapacitor. This sup-ercapacitor is modeled by the serial combination of an ESR: RS

and a complex pore impedance Zp. The pore impedance consistsof (i) two RC circuits with four elements, R1, R2, C1 and C2; and(ii) an equivalent capacitance: C0. The frequency-domain expres-sion for Zp(jx) is

ZpðjxÞ ¼s cothð

ffiffiffiffiffiffiffiffiffijxs

pÞ

C ffiffiffiffiffiffiffiffiffijxs

p ð1Þ

where x is the operation frequency; s is a correction constant, andC is the capacitance value. To successfully match the measuredspectra, the resistance RS, capacitance CSC and the time constant smust be determined. The inverse transformation of Zp(jx) was de-rived and the frequency domain and time-domain transformationsof Zp are expressed by:

Frequency Domain :k1ffiffiffiffiffiffijx

p cothk2

k1

ffiffiffiffiffiffijx

p

Time Domain :k2

1

k2þ 2k2

1

k2

X1n¼1

eðn2p2k21=k2

2Þtð2Þ

where k1, k2 are modeling constants, and n represents the numberof RC circuits in the equivalent system. The general transformationof a RC circuit is:

Frequency Domain :Ri

jxRiCi

Time Domain :1Ci exp t

RiCi

ð3Þ

Comparing coefficients from (1)–(3), parameters can be identi-fied as:

Table 1System parameters of the 350 F supercapacitor.

Symbol Meaning Value

RS The series resistance of supercapacitor 0.008 XC0 The series capacitance of supercapacitor 350 FR1 The resistance of first RC circuit 0.0012 XR2 The resistance of second RC circuit 0.00028949 XC1 The capacitance of first RC circuit 175 FC2 The capacitance of second RC circuit 175 F

Fig. 6. A unified bond graph approach for the SC modeling.

340 C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345

Capacitor Set : C0 ¼k2

k21

¼ CSC and C1 ¼ C2 ¼k2

2k21

¼ CSC

2ð4Þ

Resistance Set : R1 ¼4

p2CSC; R2 ¼

1p2CSC

since Rn

¼ 2 k2

n2p2 ¼2 s

n2p2 CSCð5Þ

With given parameters CSC and RS of the Maxwell BCAP350, andbased on Eqs. (4) and (5), Table 1 lists the values of all parametersin the equivalent electric circuit. Using Kirchhoff’s Voltage Law tovalidate Fig. 5, the voltages of the SC circuit in the frequency domaincan be expressed by

VSCðsÞ ¼ VS þ VC þ V1 þ V2

¼ iRþ iC0sþ iR1

R1C1sþ 1þ iR2

R2C2sþ 1ð6Þ

where VSC is the overall voltage across the SC, VC is the voltageacross C0, and V1 and V2 are terminal voltages across the firstand second RC circuits. Using the inverse Laplace transform tech-nique, the dynamics of the circuit in time domain can be ex-pressed by

VSCðtÞ ¼ iðtÞRS þ1C0

ZiðtÞdt þ VC0ð0Þ

þ iðtÞR1 þ ðVC1 ð0Þ iðtÞR1Þet

R1C1

n o

þ iðtÞR2 þ ðVC2 ð0Þ iðtÞR2Þet

R2C2

n oð7Þ

where i(t) represents the current passing through the main circuit.By substituting R1, R2, C0, C1, and C2 in terms of CSC as derived in (4)and (5), the analytical solution of the SC system, from Eq. (7), can bere-written as

VSCðtÞ ¼ iðtÞRS þ1

CSC

ZiðtÞdt þ VC0 ð0Þ

þ 4iðtÞp2CSC

þ VC1 ð0Þ 4iðtÞp2CSC

ep2 t

2

þ iðtÞp2CSC

þ VC2 ð0Þ iðtÞ

p2CSC

e2p2

¼ f ðCSC; iðtÞ; tÞ ð8Þ

where VSC(t) is the output and i(t) is the input. To investigate the lin-ear performance of C0, C1, and C2, which governs the third-orderdynamics, it is necessary to derive standard state-space equationsin the time domain. Let the vector of three states beX0 ¼ ½X1 X2 X3 ½C0VC C1V1 C2V2 , the system single inputu i(t) and the system single output y = VSC(t). Applying a unifiedbond graph approach [18,19], which is widely applied in multi-dis-ciplinary domains, the linear time-invariant (LTI) single input singleoutput (SISO) system can be plotted as Fig. 6. According to AppendixA, a third-order state-space equation can be derived:

_XðtÞ ¼

_X1ðtÞ_X2ðtÞ_X3ðtÞ

2664

3775 ¼

0 0 0

0 1C1R1

0

0 0 1C2R2

2664

3775

X1ðtÞX2ðtÞX3ðtÞ

264

375þ

1

1

1

264

375iðtÞ ¼ AX þ BU

yðtÞ ¼ 1C0

1C1

1C2

h i X1ðtÞX2ðtÞX3ðtÞ

264

375þ RS iðtÞ ¼ C X þ Du

ð9Þ

From Eq. (4) and (5), Eq. (9) can be represented as:

_XðtÞ ¼0 0 0

0 p2

2 0

0 0 2p2

264

375

X1ðtÞX2ðtÞX3ðtÞ

264

375þ

1

1

1

264

375iðtÞ ¼ f ðiðtÞÞ

yðtÞ ¼ 1CSC

1 2 2½ X1ðtÞX2ðtÞX3ðtÞ

264

375þ RS iðtÞ ¼ f ðiðtÞ;RS;CSCÞ

ð10Þ

Appendix A details the above derivation. From Eq. (10), the sys-tem output (performance) depends on the input drawn current, thevalue of the capacitance, and the ESR value. Clearly, the system islinear time invariant if all parameters in matrix A and matrix Bare constants (RS and CSC are constants). Hence, Eqs. (8)–(10) char-acterize the LTI supercapacitor dynamics.

The initial conditions of these three states are assumed to bezero due to the fact that capacitors store no energy at the initialstage. Thus, ½X1ð0Þ X2ð0Þ X3ð0Þ ¼ ½ 0 0 0 . The transfer func-tion TF(s), according to Eq. (9) and Appendix B, can be derived asfollows:

TFðsÞ ¼ a1s2 þ a2sþ a3

C1C2R1R2s3 þ ðC1R1 þ C2R2Þs2 þ sþ RS ð11Þ

where

a1 ¼R1R2ðC0C1 þ C1C2 þ C0C2Þ

C0

a2 ¼C1R1 þ C2R2

C0þ R1 þ R2

a3 ¼1C0

ð12Þ

Note that if the system is a linear time varying system, the val-ues of two key parameters CSC and RS vary with elapsed time. Theyare

CSC ¼ CSCðtÞRS ¼ RSðtÞ

ð13Þ

3.2. Nonlinear SC modeling in time domain

The LTI model in Section 3.1 is not so practical due to somephysical assumptions [20]: (i) one dimensional, constant transportproperties in the solution phase; (ii) concentration gradients in thesolution phase are ignored; (iii) temperature changes within thecell are ignored, etc. In real operation, however, complex features

Fig. 7. The three-layer ANN parameter identifier: (a) construction and the back-propagation training procedures; (b) training error with respect to epochs.

Fig. 8. The structure of the on-line nonlinear supercapacitor model.

C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345 341

and mutual effects cause the SC to become a highly nonlinearsystem. According to the experimental data in Section 2 and Eq.(6), capacitance can be written as a function of the three statesin Eq. (9):

CSC ¼ CSCðT;VSCðVC ;V1;V2; ÞÞ ¼ CSCðT;X1;X2;X3Þ ð14Þ

Similarly, the ESR: RS can be represented as:

RS ¼ RSðT;VSCðVC ;V1;V2; ÞÞ ¼ RSðT;X1;X2;X3Þ ð15Þ

Substituting RS and CSC in Eqs. (14) and (15) into Eq. (9), which isalso based on Eqs. (4) and (5), the LTI system become a nonlinearsystem as below:

_XðtÞ ¼

_X1ðtÞ_X2ðtÞ_X3ðtÞ

264

375 ¼

0 0 00 1

C1R10

0 0 1C2R2

2664

3775

X1ðtÞX2ðtÞX3ðtÞ

264

375þ

111

264

375iðtÞ ¼ f1ðX; UÞ

yðtÞ ¼ 1C0

1C1

1C2

h i X1ðtÞX2ðtÞX3ðtÞ

264

375þ RS iðtÞ ¼ f2ðX; UÞ

ð16Þ

where f1 and f2 denote the nonlinearity of the state-space equationsand the output equations. The superscript for parameters in Eq.(16) represents the nonlinear form, as summarized in Table 2. Byregarding the nominal values of RS and C0 as 100%, this table usesfunctions u1 and u2 to represent the nonlinear gains varying withthe operating temperature T and the supercapacitor voltage VSC.

To develop an on-line model, this study uses a three-layer ANNshown in Fig. 7a as the on-line parameter identifier. This identifierconsists of an input layer, a hidden layer, and an output layer. Thetwo inputs for the input layer are the operation temperature, T,and the overall voltage of the supercapacitor, VSC. The two outputsfor the output layers are u1(T, VSC) and u2(T, VSC), respectively. Totrain the weights between the input layers and the hidden layer(denoted by xij, i = 1, 2; j = 1, 2, . . . , 42) and the weights betweenthe hidden layer and the output layer (denoted byxjk, j = 1, 2, . . . , 42; k = 1, 2), the measured data plotted in Fig. 2and Fig. 3 should be built up as 42 input–output pairs (7 tempera-tures multiplied by 6 voltages) to feed the ANN. The calculated costand the error signal were then sent to a back-propagation algo-rithm. The weights, xij and xjk, were modified and converged to-ward certain values with a minimum system error after sufficientiterations (epochs). Fig. 6b illustrates the descent of the system er-ror, which indicates that the ANN has been completely trained. Adetailed back-propagation algorithm can be found in the literaturesource [21].

The on-line nonlinear model of the supercapacitor can be com-bined with the LTI plant in Section 3.1, as shown in Fig. 8. In thebeginning, the system input, loading current i(t), was fed to thesupercapacitor model. The variables VSC(t) and T(t) were assignedto the ANN parameter identifier at each time step. The nonlineargains, /1(t) and /2(t), were then calculated and delivered to the

Table 2Nonlinear relationships of parameters in the equivalentelectric circuit.

Symbol Nonlinear relationship

RS Rs = Rs u1(T, VUC)C0 C0 ¼ CUC u2ðT;VUCÞR1 R1 ¼ 4s

p2C0¼ 4s

p2ðC0u1u2Þ

R2 R2 ¼ 16sp2C0¼ 8s

p2ðC0u1u2Þ

C1 C1 ¼ 0:5CUC ¼ 0:5CUC u1 u2

C2 C2 ¼ 0:5CUC ¼ 0:5CUC u1 u2

model to determine the parameters in Table 2. The single outputVSC was finally derived precisely.

4. Simulation results and discussion

4.1. Simulation of the LTI supercapacitor model

This section discusses the performance of the linear time invari-ant model of the supercapacitor described in Section 3.1. Based onEq. (11) and the values of parameters in Table 1, the transfer func-tion TF(s) for the Maxwell BCAP 350 can be derived as

TFðsÞ ¼ VSCðsÞiðsÞ ¼

0:008s3 þ 0:2103s2 þ 0:9619sþ 0:2686sðs2 þ 24:5Sþ 93:99Þ ð17Þ

Fig. 9. Comparison of the frequency response between the measured data and the LTI model.

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

Elapsed Time (s)

Volta

ge (V

)

Experimental Data at C. C. 13.0AExperimental Data at C. C. 16.5AExperimental Data at C. C. 20.0ALTI Simulation at C. C. 13.0ALTI Simulation at C. C. 16.5ALTI Simulation at C. C. 20.0A

10 15 20 25 300.5

1

1.5

2.5V Loss of Nonlinearity

Fig. 10. Comparison of experimental data and simulation results of the LTI model inconstant current mode at 13.0 A, 16.5 A, and 20.0 A.

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

Elapsed Time (s)

Volta

ge (V

)

Experimental Data at C. C. 13.0AExperimental Data at C. C. 16.5AExperimental Data at C. C. 20.0ANonlinear Simulation at C. C. 13.0ANonlinear Simulation at C. C. 16.5ANonlinear Simulation at C. C. 20.0A

Fig. 11. Comparison of experimental data and simulation results of the nonlinearmodel in constant current mode at 13.0 A, 16.5A, and 20.0 A.

342 C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345

The LTI model is a third-order system with a pole located at zeroand a pair of complex conjugate poles represented by the last termin the denominator. Fig. 9 displays the Bode diagram after deliver-ing the frequency range of interest from 0.01 Hz to 100 Hz to excitethe linear system. The magnitude of the output/input gain, Y(s)/U(s) = VSC(s)/i(s), is 27 dB at 0.01 Hz. The gain curve decreaseswith a slope of 45 due to the single pole at s = 0 in Eq. (17).The phase lag is 80 initially, and increases as the frequency rises.This is because the effects from the capacitances which governthe dynamics at low frequency, leading to a greater phase lag.The influence of RS is stronger at higher frequencies, causing thephase lag to decrease at this range. This figure reveals that the

bandwidth, defined as 0.707 max (Y(s)/u(s)), is within the properoperation range. That means the system performances well below0.016 Hz (62.5 seconds per charge/discharge cycle). With a highergain of VSC/i at low frequencies, the supercapacitor is able to storemore electrical energy. Too fast charge or discharge will causeinsufficient energy stored or released. The magnitude reaches to42 dB and the phase lag 0 at the maximum of the frequencyinterested.

In the time-domain simulation, the DC power supplier providesthree constant currents of 13.0 A, 16.5 A, and 20.0 A to charge thesupercapacitor, respectively. Once the VSC reaches 2.5 V, the super-visory personal computer regards the SC as being fully charged andthe relays cut off the electricity so that the current returns to zero.Fig. 10 compares the simulation results of the LTI model with the

0 20 40 60 80 100250

300

350

400C

o (F)

Co in LTI System at C. C. 16.5A

Co in Nonlinear System at C. C. 16.5A

0 20 40 60 80 1006

8

10

12 x 10-3

Elapsed Time (s)

Rs (Ω

)

Rs in LTI System at C. C. 16.5A

Rs in Nonlinear System at C. C. 16.5A

Fig. 12. On-line simulation results of two key parameters: C0 and RS.

C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345 343

measured data, indicating that VSC is initially proportional to thecharging current due to the constant RS.

When the charging process is completed at the 40th second, 48thsecond, and 63rd second for the cases of charging current at 20 A,16.5 A and 13 A, respectively, the voltages suddenly drops becausei(t) is set to zero at then. The voltage drops, similar to the initial con-dition, are proportional to the charging currents because they havethe same RS (with the same operation conditions: VSC = 2.5 V,T = 300 K). Although this case study with voltage overshoot is notso practical in the realistic operation, however, it clearly revealsthe error dynamics due to ignorance of the nonlinear effects.

0 5 10 15 200

0.5

1

1.5

2

2.5

3

Elapsed Time (s)

Volta

ge (V

)

Experimental Data at 270K(3A)Experimental Data at 330K(3A)Nonlinear Simulation at 270K(3A)Nonlinear Simulation at 330K(3A)

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

Elapsed Time (s)

Volta

ge (V

)

Experimental Data at C. R. 0.5ΩExperimental Data at C. C. 3.0ANonlinear Simulation at C. R. 0.5ΩNonlinear Simulation at C. C. 3.0A

(a) (

(c) (

Fig. 13. Online charging/discharging simulation and experiment of Maxwell PC10 at: (a)different combinations of two supercapacitors.

4.2. Simulation of the nonlinear supercapacitor model

The second case study describes the simulation of the proposedon-line nonlinear model with the three-layer ANN parameter iden-tifier. Fig. 11 compares the simulation results with the experimentdata. With the same testing conditions as above, the simulation re-sults and experimental data are in excellent agreement. The non-linear model is much better than the LTI model due to the on-line tuning of RS and C0 in the model.

Fig. 12 shows the online simulation results of the two keyparameters: RS and C0, using linear and nonlinear models with con-stant current charge at 16.5 A. In the LTI simulation, C0 and RS areconstant (C0 = 350 F, RS = 0.008 X), the effects from the environ-ment is not considered. For the nonlinear case, the on-line ANNparameter identifier evaluates C0 and RS at each sampling, whichwas set to 0.1 second per sample. The capacitance C0 is 260 F ini-tially when VSC = 0 V. As the charging process is activated, C0 in-creases with the increasing of VSC. It reaches 352 F whenVSC = 2.5 V at the 48th second. Meanwhile, C0 slightly decreasesdue to a 0.15 V drop of VSC and i = 0 A. For the nonlinear RS simula-tion, the variation of the curve exhibits an opposite tendency com-pared to the nonlinear C0 case. According to Fig. 3a, RS decreases asVSC increases, the value of RS finally converges to the nominal va-lue, 8 103 X.

Fig. 13 illustrates another example to prove the generality ofthis on-line nonlinear modeling technique. This case study simu-lates the charging and discharging performance of a MaxwellPC10 supercapacitor with a nominal capacitance 10 F. The simula-tion was carried out by setting up the transfer function and state-space equations from the LTI model, and then constructing the

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Elapsed Time (s)

Volta

ge (V

)

Experimental Data at C. C. 1AExperimental Data at C. C. 2AExperimental Data at C. C. 3ANonlinear Simulation at C. C. 1ANonlinear Simulation at C. C. 2ANonlinear Simulation at C. C. 3A

0 5 10 15 20 250

1

2

3

4

5

Elapsed Time (s)

Volta

ge (V

)

Experimental Data in Series(3A)Experimental Data in Series(3A)Nonlinear Simulation in Parallel(3A)Nonlinear Simulation in Parallel(3A)

b)

d)

different currents, (b) different temperatures, (c) different discharging rates, and (d)

344 C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345

nonlinear dynamic model using the three-layer ANN parameteridentifier. Fig. 13a compares the results of the nonlinear modelwith the experimental data at three constant current charges:1 A, 2 A, and 3 A. This figure shows that the resulting profilesmatch well. Fig. 13b examines the influence of different operationtemperatures at 270 K and 300 K under constant current 3 A, indi-cating that simulation and measurement agree well. When i = 0,the voltage drops in the case of 270 K is greater than that at300 K. This is because the system, at VSC = 2.5 V, has a larger RS atlower temperature as shown in Fig. 3a. Fig. 13c displays the simu-lation and experimental results at constant current mode (3A) andconstant resistance mode (0.5 X) in during the discharging pro-cess. The initial VSC is set at 2 V, while the activation time is setat the 1st second. Similar to the results in the charging process,the simulation results in both modes track the measured data well.Fig. 13d simulates two supercapacitors connected in series and inparallel. The excellent agreement between the simulation andthe experiment confirms that this nonlinear modeling techniquecan be extended to various combinations of multi-supercapacitors.

5. Conclusions

This paper successfully establishes and verifies the linear andnonlinear models of two types of supercapacitors. A third-orderequivalent electric circuit was set up based on the theory of theEIS and the AC impedance tests. To model the LTI system, we em-ployed a unified bond graph approach to derive the transfer func-tion and the state-space equations of the dynamic system. We alsoidentified the parameters of those six components in the electriccircuit. For the nonlinear model, a three-layer artificial neural net-work was employed to generate the highly nonlinear relationshipsbetween operation temperature, voltage, and those unknownparameters in the dynamic equations. Simulation results show thatthe LTI model is able to analyze the working area in the frequencydomain from the Bode diagram. In the time domain, the transientsimulation shows the disadvantage of the LTI model that lacks ofnonlinear effects. As the on-line nonlinear modeling technique,the trained ANN and the back-propagation algorithm are able totrack the nonlinear effects precisely. Comparing with the experi-mental results, on-line simulation results show that the accuracyof the nonlinear model is much better than that of the linear mod-el. This on-line nonlinear supercapacitor model will be integratedwith batteries and fuel cells in the near future for developing theenergy management system and control strategy of a hybrid elec-tric vehicle.

Acknowledgments

The authors would like to thank the National Science Council,Taiwan, for financially supporting this research under contractsof NSC 93-2218-E-007-050, NSC 94-2218-E-007-025 and NSC 95-2218-E-007-016. The scholarship to Mr. C.H. Wu is gratefullyacknowledged.

Appendix A. Derivation of a third-order linear single inputsingle output (SISO) supercapacitor model

To derive system dynamics equations, a unified bond graph ap-proach has been adopted here. This approach is able to generate acorresponding graphical system efficiently, and the mathematicalequations can be derived using those constitutive laws. The bondgraph of the SC equivalent system can be plotted in Fig. 6. Forthe SC set, the state-space equations are described as below:

(i) System States: X’ ¼ ½X1 X2 X3 ½C0VC C1V1 C2V2 ,

(ii) System input u i(t)(iii) System output y = VUC(t)

According to the numbered bonds in the bond graph, the systemstates are:

X1 ¼ C0e4

X2 ¼ C1e7

X3 ¼ C2

ðA:1Þ

After differentiating Eq. (A.1) and following the constitutivelaws, Kirchhoff’s voltage laws, and Kirchhoff’s circuit laws, the dy-namic equations of those capacitors in the bond graph can be ex-pressed by:

_X1 ¼ C0 _e4 ¼ f4 ¼ iðtÞ ¼ u

_X2 ¼ C1 _e7 ¼ f7 ¼ f5 f6 ¼ u X2

C1R1

_X3 ¼ C2 _e9 ¼ f9 ¼ f3 f8 ¼ u X3

C2R2

ðA:2Þ

where symbols e, f are defined as ‘‘effort’’ and ‘‘flow’’ in the bondgraph. The suffix number represents the bond number correspond-ingly. From Eq. (A.2), the dynamic equations can be re-written as astandard state-space form as shown in Eq. (9). Finally, from Eq. (6)and (A.1), the control equation can be expressed by:

y ¼ VUCðtÞ ¼ VS þ VC þ V1 þ V2 ¼ u RS þX1

C0þ X2

C1þ X3

C2

¼ 1C0

1C1

1C2

h i X1

X2

X3

264

375þ RS u ðA:3Þ

Appendix B. Derivation of a transfer function for thesupercapacitor

Equation (9) is in the form of a standard state-space equation.To transfer it into the frequency domain, the procedures are de-scribed as below.

Assume that the initial conditions of those three states arezeros: Xð0Þ ¼ 0 (i.e. no electricity stored into the capacitors). Thestate-space equation can be transferred into the frequency domainby the Laplace transformation. The equations are re-written asbelow:

Time Domain:

XðtÞ ¼ AXðtÞ þ BuðtÞyðtÞ ¼ CXðtÞ þ DuðtÞ

ðB:1Þ

Laplace Transform:

sXðsÞ ¼ AXðsÞ þ BUðsÞ ðB:2Þ

YðsÞ ¼ CXðsÞ þ DUðsÞ ðB:3Þ

Eq. (B.2) can be re-written by:

XðsÞ ¼ ðsI AÞ1BUðsÞ ðB:4Þ

Substitute Eq. (B.4) into Eq. (B.3), the input–output relationship canbe derived. It is

YðsÞ ¼ CðsI AÞ1BUðsÞ þ DUðsÞ ¼ ½CðsI AÞ1Bþ DUðsÞ TFðsÞUðsÞ ðB:5Þ

The transfer function TF(s) is

TFðsÞ ¼ CðsI AÞ1Bþ D ¼ CadjðsI AÞBjsI Aj þ D ðB:6Þ

C.H. Wu et al. / Energy Conversion and Management 53 (2012) 337–345 345

where the matrices A–D are from Eq. (9), and Eq. (11) can bederived.

References

[1] Conway BE. Electrochemical supercapacitors: scientific fundamentals andtechnological applications. New York: Kluwer Academic/Plenum Publishers;1999.

[2] Adib E, Farzanehfard H. Soft switching bidirectional DC–DC converter forultracapacitor–batteries interface. Energy Convers Manage 2009;50:2879–84.

[3] Ayad MY, Pierfederici S, Rael S, Davat B. Voltage regulated hybrid DC powersource using supercapacitors as energy storage device. Energy Convers Manage2007;48:2196–202.

[4] Andrzej L, Maciej G. Practical and theoretical limits for electrochemical double-layer capacitors. J Power Sources 2007;173:822–8.

[5] Spyker RL, Nelms RM. Classical equivalent circuit parameters for a double-layer capacitor. IEEE Trans Aerospace Elect Syst 2000;36:829–36.

[6] Gualous H, Bouquain D, Berthon A. Experimental study of supercapacitor serialresistance and capacitance variations with temperature. J Power Sources2003;123:86–93.

[7] Buller S, Karden E, Kok D, De Doncker RW. Modeling the dynamic behavior ofsupercapacitors using impedance spectroscopy. IEEE Trans Indus Appl2002;38:1622–6.

[8] Michel H. Temperature and dynamics problems of ultracapacitors in stationaryand mobile applications. J Power Sources 2006;154:556–60.

[9] Lajnef W, Vinassa J-M, Briat O, Azzopardi S, Woirgard E. Characterizationmethods and modelling of ultracapacitors for use as peak power sources. JPower Sources 2007;168:553–60.

[10] Rafik F, Gualous H, Gallay R, Crausaz A, Berthon A. Frequency, thermal andvoltage supercapacitor characterization and modeling. J Power Sources2007;165:928–34.

[11] Shi L, Crow ML. Comparison of ultracapacitor electric circuit models. IEEEPower Energy Soc General Meet 2008;8:1–6.

[12] Sakka MAI, Gualous H, Van Mierlo J, Culcu H. Thermal modeling and heatmanagement of supercapacitor modules for vehicle applications. J PowerSources 2009;194:581–7.

[13] Brouji HEI, Briat O, Vinnasa JM, Henry H, Woirgard E. Analysis of the dynamicbehavior changes of supercapacitors during calendar life test under severalvoltages and temperatures conditions. Microelect Reliab 2009;49:1391–7.

[14] Bhatikar S, Mahajan RL, Wipke KB, Johnson V. Artificial neural network basedenergy storage system modeling for hybrid electric vehicles. In: Proceeding ofthe Future Car Congress; 2000.

[15] Shen WX, Chan CC, Lo EWC, Chau KT. A new battery available capacityindicator for electric vehicles using neural network. Energy Convers Manage2002;43:817–26.

[16] Abolhassani Monfared N, Gharib N, Moqtaderi H, Hejabi M, Amiri M, et al.Prediction of state-of-charge effects on lead-acid battery characteristics usingneural network parameter modifier. J Power Sources 2006;158:932–5.

[17] Farsi H, Global F. Artificial neural network simulator for supercapacitorperformance prediction. Comput Mater Sci 2007;39:678–83.

[18] Hung YH, Lin PH, Wu CH, Hong CW. Real-time dynamic modeling of hydrogenPEMFCs. J Franklin Inst 2008;345:182–203.

[19] Hung YH, Hong CW. Bond graph modeling of fuel cell and engine hybridelectric scooters. Int J Vehicle Des 2007;45:522–41.

[20] Skha G, White RE, Popov BN. A mathematical model for a lithium-ion battery/electrochemical capacitor hybrid system. J Electrochem Soc2005;152:1682–93.

[21] Haykin S. Neural Networks: a comprehensive foundation. 2nd ed. NewJersey: Prentice-Hall; 1999.